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Seems funny that Turing “…assumed that machines could not operate
with infinite numbers…” given that the tape is assumed to be infinite.

Not really infinite but it has no boundaries, it can always extend
if needed. At any given time the used tape is of finite length.

Right. The tape is not part of the machine. It plays the role of an
extendible environment/memory space.

If we had infinite tape ourself, we would never have drawn on cave's
wall, nor invented magnetic tapes.

You can always substitute the message "memory overflow" by "give me
more memory space".

True, the Turing machine has an infinite tape, but that is pedagogical
folklore. To be finite is part of the essence of what a machine
consists in, as Turing was well aware off. For example by proving the
equivalence Turing-machine with lambda term (which have no tape at all).

Also, machine can handle the infinite numbers. It can handle all the
constructive infinite ordinal in an effective way, like us, and it can
handle the others infinite ordinals and cardinals in a non
constructive way, like us.

Maybe not. In Turing's proof he assumed that machines could not
operate with infinite numbers, so if there is a theory of everything
(and there might not be) and if you know it and if you can use
nothing but that to show independently of Turing that no machine can
solve the Halting Problem then that would prove that irrational
numbers with a infinite number of digits play no part in the
operation of the universe; on the other hand if this new physical
theory shows you how to make such a machine then we'd know that
nature understands and uses infinity. I admit that I used the word "
if " a lot in all that.

John K Clark
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